Quantum optical CNOT gate

ABSTRACT

A nondeterministic quantum CNOT gate ( 10 ) for photon qubits, with success probability 1/9, uses beamsplitters (B 1 –B 5 ) with selected reflectivities to mix control and target input modes. It may be combined with an atomic quantum memory to construct a deterministic CNOT gate, with applications in quantum computing and as a Bell-state analyser.

TECHNICAL FIELD

This invention concerns a non-deterministic quantum CNOT gate for photonqubits. It may be used in a deterministic CNOT gate. Such adeterministic CNOT gate would find many applications in QuantumInformation Technology including as a Bell-state analyser and as a keyelement in an Optical Quantum Computer.

BACKGROUND ART

Classically, information can be encoded digitally by using a collectionof systems which can each be placed in one of two distinct states,labelled as zeros and ones. Each system carries one “bit” ofinformation. On the other hand quantum mechanics has systems that can beplaced in states, called “superpositions”, that exist simultaneously aszeros and ones. For example information could be encoded with singlephotons by labelling a horizontally polarized photon, |H>, as a one anda vertically polarized photon, |V>, as a zero. But a photon can also bein a superposition of these two states such as 1/√{square root over(2)}(|H>+|V>). Such a state carries one “qubit” of quantum information[1]. Furthermore quantum systems can be “entangled”. Entanglement refersto correlation between the measured properties of sub-systems thatcannot be explained in terms of classical correlation.

These different properties of quantum information make new technologiespossible, in particular “quantum cryptography” [2] which uses quantuminformation to distribute cryptographic keys with near perfect securityand “quantum computing” [3] which utilizes the increased processingpower of quantum information to dramatically reduce the time requiredfor certain calculations. Whilst photons are easily manipulated at thesingle qubit level, long distance quantum cryptography and all quantumcomputing requires two qubit gates. Two qubit gates are not easilyimplemented in optics.

A key two qubit gate is the Controlled Not (CNOT) gate. It operates inthe following way: One qubit is the “control” and the other qubit is the“target”. The control qubit emerges from the gate in the same logicalstate as it entered. If the control qubit is in the logical zero statethen the target qubit emerges with the same state as it entered. If thecontrol qubit is in the logical one state then the target qubit emergeswith the opposite state to that with which it entered the gate. If thecontrol qubit is in a superposition state then the control and targetqubits become entangled.

DISCLOSURE OF THE INVENTION

The invention is a non-deterministic quantum CNOT gate for photonqubits, comprising an interferometer to receive two target photon modesand two control photon modes and to cause a sign shift by splitting andremixing of the target photon modes, conditional on the presence of aphoton in one particular control photon mode, so that the target photonqubit swaps modes if the control quantum qubit is in one mode but doesnot if the control photon qubit is in the other mode, provided acoincidence is measured between the control and target output photonmodes.

The interferometer may comprise five beamsplitters. A first beamsplitterto receive two target photon modes. A second beamsplitter to receive oneoutput mode from the first beamsplitter and a control photon mode, andto deliver a control output mode. A third beamsplitter to receive theother output mode from the first beamsplitter. A fourth beamsplitter toreceive the outputs of the second and third beamsplitters and deliverthe target output modes. And a fifth beamsplitter to receive the othercontrol photon mode, and deliver the other control output mode. Thebeamsplitters are asymmetric in phase.

Reflection off the bottom of the first, third and fourth beamsplittersmay produce sign change, but reflections off the top of the second andfifth beamsplitters may produce a sign change.

The first and fourth beamsplitters may both be 50:50 (η=0.5). Thesecond, third and fourth beamsplitters may have equal reflectivities ofone third (η=0.33).

The Heisenberg equations relating the control and target input modes tothe their corresponding outputs may be:

$\begin{matrix}\begin{matrix}{c_{Ho} = {\frac{1}{\sqrt{3}}\left( {c_{H} + {\sqrt{2}\mspace{11mu} v_{4}}} \right)}} \\{c_{Vo} = {\frac{1}{\sqrt{3}}\left( {{- c_{V}} + {\sqrt{2}\mspace{11mu} t^{\prime}}} \right)}} \\{t_{Ho} = {\frac{1}{\sqrt{2}}\left( {t^{''} + t^{\prime''}} \right)}} \\{t_{Vo} = {\frac{1}{\sqrt{2}}\left( {t^{''} - t^{''\prime}} \right)}} \\{{where}\text{:}} \\{t^{''} = {\frac{1}{\sqrt{3}}\left( {t^{\prime} + {\sqrt{2}c_{v}}} \right)}} \\{t^{''\prime} = {{\frac{1}{\sqrt{6}}\left( {t_{H} - t_{v}} \right)} + {\sqrt{\frac{2}{3}}\; v_{5}}}} \\{t^{\prime} = {\frac{1}{\sqrt{2}}\left( {t_{H} + t_{v}} \right)}}\end{matrix} & (2)\end{matrix}$

By “non-deterministic” we mean it operates with a finite successprobability. It is constructed from linear optics (as opposed tonon-linear optics [4]) and is significantly simpler than previoussuggestions [5].

When combined with verifiable quantum memory (VAM), thenon-deterministic quantum CNOT gate may form a deterministic (that is italways works) CNOT gate using the teleportation protocol.

The general properties required of a VAM are as follows:

-   -   (i) The quantum state of a photon can be transferred onto an        atomic quantum state, held with minimal decoherence for some        time and then reconverted to a photon quantum state efficiently.    -   (ii) Probing of another atomic transition can verify that the        photon qubit has been placed into memory.    -   (iii) Joint measurements of the Bell type can be implemented        between two qubits held in atomic memory.

The deterministic CNOT gate may also comprise two spontaneousdown-converters to produce pairs of polarization entangled photons whichare stored in four VAMs. Two of the photons are released and sentthrough the non-deterministic quantum CNOT gate and they aresubsequently stored in two further VAMs. Polarization rotations areperformed conditional on the results of the Bell measurements to achievethe required functionality.

In more detail, the photons in the second and third VAM may be releasedand sent through the non-deterministic quantum CNOT gate, and they arethen stored in the two further VAMs. And the photons stored in thefirst, fifth, sixth and fourth VAMs are in the state required toimplement the teleportation protocol [7].

The control and target photon qubits enter and may be stored in aseventh and eighth VAM. Bell measurements are made on the stored qubitsin the first and seventh VAMs and on the eighth and fourth VAMs. Thequbits in fifth and sixth VAMs are released. Polarization rotations ofσ_(X)≡X and σ_(Z)≡Z are performed on the qubit released from the fifthVAM conditional on the results of the Bell measurements on the first andseventh VAMs and on the eighth and fourth VAMs as indicated in thefigure. Similarly rotations are performed on the photon released fromthe sixth VAM dependent on the results of the Bell measurements.

BRIEF DESCRIPTION OF THE DRAWINGS

An example of the invention will now be described with reference to theaccompanying drawings, in which:

FIG. 1 is a schematic diagram of a non-deterministic quantum CNOT gatefor photon qubits, where dashing indicates the surface from which a signchange occurs upon reflection; and

FIG. 2 is a schematic diagram of a deterministic CNOT gate.

BEST MODES OF THE INVENTION

1. Non-Deterministic Quantum CNOT Gate

Referring first to FIG. 1, the non-deterministic quantum CNOT gate 10comprises five beamsplitters, B1, B2, B3, B4, and B5 which are allassumed asymmetric in phase. That is, it is assumed that the operatorinput/output relations (the Heisenberg equations) between two input modeoperators (a_(in) and b_(in)) and the corresponding output operators(a_(out) and b_(out)) for the beamsplitters have the general form:

$\begin{matrix}\begin{matrix}{a_{out} = {{\sqrt{\eta}\mspace{11mu} a_{i\; n}} + {\sqrt{1 - \eta}\mspace{11mu} b_{i\; n}}}} \\{b_{out} = {{\sqrt{1 - \eta}\mspace{11mu} a_{i\; n}} - {\sqrt{\eta}\mspace{11mu} b_{i\; n}}}}\end{matrix} & (1)\end{matrix}$where η (1−η) is the reflectivity (transmitivity) of the beamsplitter.Reflection off the bottom produces the sign change except for B2 and B4which have a sign change by reflection off the top. This phaseconvention simplifies the algebra but other phase relationships willwork equally well in practice. Beamsplitters B1 and B3 are both 50:50(η=0.5). The beamsplitters B2, B4 and B5 have equal reflectivities ofone third (η=0.33).

We employ dual rail logic such that the “control in” qubit isrepresented by the two bosonic mode operators c_(H) and c_(V). A singlephoton occupation of c_(H) with c_(V) in a vacuum state will be ourlogical 0, which we will write |H> (to avoid confusion with the vacuumstate). Whilst a single photon occupation of c_(V) with c_(H) in avacuum state will be our logical 1, which we will write |V>.Superposition states can also be formed via beamsplitter interactions.Similarly the “target in” is represented by the bosonic mode operatorst_(H) and t_(V) with the same interpretations as for the control.

The Heisenberg equations relating the control and target input modes totheir corresponding outputs are

$\begin{matrix}\begin{matrix}{c_{Ho} = {\frac{1}{\sqrt{3}}\left( {c_{H} + {\sqrt{2}\mspace{11mu} v_{4}}} \right)}} \\{c_{Vo} = {\frac{1}{\sqrt{3}}\left( {{- c_{V}} + {\sqrt{2}\mspace{11mu} t^{\prime}}} \right)}} \\{t_{Ho} = {\frac{1}{\sqrt{2}}\left( {t^{''} + t^{\prime''}} \right)}} \\{t_{Vo} = {\frac{1}{\sqrt{2}}\left( {t^{''} - t^{''\prime}} \right)}} \\{{where}\text{:}} \\{t^{''} = {\frac{1}{\sqrt{3}}\left( {t^{\prime} + {\sqrt{2}c_{v}}} \right)}} \\{t^{''\prime} = {{\frac{1}{\sqrt{6}}\left( {t_{H} - t_{v}} \right)} + {\sqrt{\frac{2}{3}}\; v_{5}}}} \\{t^{\prime} = {\frac{1}{\sqrt{2}}\left( {t_{H} + t_{v}} \right)}}\end{matrix} & (2)\end{matrix}$

The gate operates by causing a sign shift in the interferometer formedby the splitting and remixing of the target modes, conditional on thepresence of a photon in the c_(V) mode. Thus the target modes swap ifthe control is in the state |V>_(c) but do not if the control is instate |H>_(c). This is always true when a coincidence (photons aredetected at the same time) is measured between the control and targetoutputs. However such coincidences only occur one ninth of the time, onaverage. The other eight ninths of the time either the target or thecontrol or both do not contain a photon. This can be seen explicitly bycalculating coincident expectation values from Eqs 2. If the initialstate is |H>_(c) |H>_(t) then we find

$\begin{matrix}\begin{matrix}{\left\langle {c_{H0}^{\dagger}c_{H0}\mspace{11mu} t_{H0}^{\dagger}\; t_{H0}} \right\rangle = \frac{1}{9}} \\{\left\langle {c_{H0}^{\dagger}c_{H0}\mspace{11mu} t_{V0}^{\dagger}\; t_{V0}} \right\rangle = 0} \\{\left\langle {c_{V0}^{\dagger}c_{V0}\mspace{11mu} t_{V0}^{\dagger}\; t_{V0}} \right\rangle = 0} \\{\left\langle {c_{V0}^{\dagger}c_{V0}\mspace{11mu} t_{H0}^{\dagger}\; t_{H0}} \right\rangle = 0}\end{matrix} & (3)\end{matrix}$and for initial state |H>_(c)|V>_(t) we find

$\begin{matrix}{\left\langle {c_{H0}^{\dagger}c_{H0}\mspace{11mu} t_{V0}^{\dagger}\; t_{V0}} \right\rangle = \frac{1}{9}} & (4)\end{matrix}$with all other expectation values zero, but for initial state|V>_(c)|H>_(t) we find

$\begin{matrix}{\left\langle {c_{V0}^{\dagger}c_{V0}\mspace{11mu} t_{V0}^{\dagger}\; t_{V0}} \right\rangle = \frac{1}{9}} & (5)\end{matrix}$with all other expectation values zero, and for initial state|V>_(c)|V>_(t) we find

$\begin{matrix}{\left\langle {c_{V0}^{\dagger}c_{V0}\mspace{11mu} t_{H0}^{\dagger}\; t_{H0}} \right\rangle = \frac{1}{9}} & (6)\end{matrix}$with all other expectation values zero. To confirm experimentally thatthe gate operates correctly requires only direct detection of the photonoutputs. For some applications this may be sufficient. However toproduce a deterministic CNOT gate from the invention requires thenon-destructive determination of whether a control and target photon areboth present. This is discussed in the following section.2. Deterministic CNOT Gate

In order to use the invention to build a deterministic CNOT gate averifiable atomic quantum memory (VAM) is required, such as thatproposed from current technology [6].

The general properties required of a VAM are as follows:

-   -   (i) The quantum state of a photon can be transferred onto an        atomic quantum state, held with minimal decoherence for some        time and then reconverted to a photon quantum state efficiently.    -   (ii) Probing of another atomic transition can verify that the        photon qubit has been placed into memory.    -   (iii) Joint measurements of the Bell type [1] can be implemented        between two qubits held in atomic memory.

In principle Ref. [6] satisfies all these criteria. If VAMs with theproperties described above are coupled with the non-deterministicquantum CNOT gate 10 then a deterministic CNOT gate can be constructedusing the teleportation protocol described in Ref. [7]. A deterministicCNOT gate 100 is shown in FIG. 2 and is described in the following:

-   (1) Entanglement Preparation: Two spontaneous down-converters, 102    and 104, (established technology [8]) produce pairs of polarization    entangled photons which are stored in VAM₁ 106, VAM₂ 108, VAM₃ 110,    and VAM₄ 112, respectively. The photons in VAM₂ 108, and VAM₃ 110,    are released and sent through the non-deterministic quantum CNOT    gate 10, and an attempt to store them in VAM₅ 114, and VAM₆ 116, is    made. Most of the time storage of photons in VAM₅ 114, and VAM₆ 116,    will not be verified (because the invention is non-deterministic) at    which point the photons in VAM₁ 106 and VAM₄ 112, are dumped and the    process is started again. However, on average one in nine attempts    will be successful and the photons stored in VAM₁ 106, VAM₅ 114,    VAM₆ 116, and VAM₄ 112, are in the state required to implement the    teleportation protocol [7].-   (2) Implementing the Gate: The photon qubits, c 130, and t 132,    enter and are stored in VAM₇ 118, and VAM₈ 120 respectively. Bell    measurements 140, 142 are made on the stored qubits in VAM₁ 106, and    VAM₇ 118, and on VAM₈ 120, and VAM₄ 112 respectively. The qubits in    VAM₅ 114, and VAM₆ 116, are released. Polarization rotations of    σ_(X)≡X and σ_(Z)≡Z (established technology) are performed on the    qubit released from VAM₅ 114 conditional on the results of the Bell    measurements 140, 142, on VAM₁ 106, and VAM₇ 118, and on VAM₈ 120,    and VAM₄ 112, as indicated in FIG. 2. Similarly rotations are    performed on the photon released from VAM₆ 116, are dependent on the    results of the Bell measurements 140 and 142.

This completes the protocol as the output qubits, c′ 150, and t′ 152,are now the original qubits with a CNOT applied.

REFERENCES

-   [1] C. H. Bennett and D. P. DiVincenzo, Nature 404, 247 (2000).-   [2] C. H. Bennett, et al, Phys Rev Lett 70, 1895 (1993).-   [3] R. P. Feynman, Foundations Phys. 16, 507 (1986); D. Deutsch,    Proc.Roy.Soc.London A400, 97 (1985).-   [4] G. J. Milburn, Phys Rev Lett 62, 2124 (1988).-   [5] E. Knill, R. Laflamme and G. J. Milburn, Nature 409, 46, (2001).-   [6] J. H. Shapiro, LANL preprint quant-ph/0105055 (2001).-   [7] D. Gottesman and I. L. Chuang, Nature 402, 390 (1999).-   [8] P. G. Kwiat, K. Mattle, H. Weinfurter and A. Zeilinger, Phys Rev    Lett 75, 4337 (1995).

It will be appreciated by persons skilled in the art that numerousvariations and/or modifications may be made to the invention as shown inthe specific embodiments without departing from the spirit or scope ofthe invention as broadly described. The present embodiments are,therefore, to be considered in all respects as illustrative and notrestrictive.

1. A non-deterministic quantum CNOT gate for photon qubits, comprisingan interferometer to receive two target photon modes and two controlphoton modes and two unoccupied ancilla modes and to cause a sign shiftby splitting and remixing of the target photon modes, conditional on thepresence of a photon in one particular control photon mode, so that thetarget photon qubit swaps modes if the control quantum qubit is in onemode but does not if the control photon qubit is in the other mode,provided a coincidence is measured between the control and target outputphoton modes.
 2. A non-deterministic quantum CNOT gate according toclaim 1, where the interferometer comprises five beamsplitters: a firstbeamsplitter to receive two target photon modes, a second beamsplitterto receive one output mode from the first beamsplitter and a controlphoton mode, and to deliver a control output mode, a third beamsplitterto receive the other output mode from the first beamsplitter, a fourthbeamsplitter to receive the outputs of the second and thirdbeamsplitters and deliver the target output modes, and a fifthbeamsplitter to receive the other control photon mode, and deliver theother control output mode.
 3. A non-deterministic quantum CNOT gateaccording to claim 2, where the beamsplitters are asymmetric in phase,reflection off the bottom of the first, third and fourth beamsplittersproduce sign change, but reflections off the top of the second and fifthbeamsplitters produce a sign change.
 4. A non-deterministic quantum CNOTgate according to claim 3, where the first and fourth beamsplitters areboth 50:50 (η=0.5), and the second, third and fourth beamsplitters haveequal reflectivities of one third (η=0.33).
 5. A non-deterministicquantum CNOT gate according to claim 2, where the first and fourthbeamsplitters are both 50:50 (η=0.5), and the second, third and fourthbeamsplitters have equal reflectivities of one third (η=0.33).
 6. Anon-deterministic quantum CNOT gate according to claim 2, 3, 5 or 4where the Heisenberg equations relating the control and target inputmodes to the their corresponding outputs are: $\begin{matrix}{c_{Ho} = {\frac{1}{\sqrt{3}}\left( {c_{H} + {\sqrt{2}\mspace{11mu} v_{4}}} \right)}} \\{c_{Vo} = {\frac{1}{\sqrt{3}}\left( {{- c_{V}} + {\sqrt{2}\mspace{11mu} t^{\prime}}} \right)}} \\{t_{Ho} = {\frac{1}{\sqrt{2}}\left( {t^{''} + t^{\prime''}} \right)}} \\{t_{Vo} = {\frac{1}{\sqrt{2}}\left( {t^{''} - t^{''\prime}} \right)}} \\{{where}\text{:}} \\{t^{''} = {\frac{1}{\sqrt{3}}\left( {t^{\prime} + {\sqrt{2}c_{v}}} \right)}} \\{t^{''\prime} = {{\frac{1}{\sqrt{6}}\left( {t_{H} - t_{v}} \right)} + {\sqrt{\frac{2}{3}}\; v_{5}}}} \\{t^{\prime} = {\frac{1}{\sqrt{2}}{\left( {t_{H} + t_{v}} \right)\;.}}}\end{matrix}$
 7. A deterministic CNOT gate comprising anon-deterministic quantum CNOT gate according to claim 1, and furthercomprising two spontaneous down-converters to produce pairs ofpolarization entangled photons which are stored in first, second, thirdand fourth verifiable quantum memorys (VAMs), where two of the storedphotons are released and sent though the non-deterministic quantum CNOTgate and are subsequently stored in fifth and sixth VAMs, andpolarization rotations are performed conditional on the results of Bellmeasurements to achieve the required functionality.
 8. A deterministicCNOT gate according to claim 7, where photons in the second and thirdVAM are released and sent though the non-deterministic quantum CNOTgate, and they are then stored in the fifth and sixth VAMs, and thephotons stored in the first, fifth, sixth and fourth VAMs are in thestate required to implement the teleportation protocol.
 9. Adeterministic CNOT gate according to claim 8, where control and targetphoton qubits enter and are stored in a seventh and eighth VAM, Bellmeasurements are made on the qubits stored in the first and seventh VAMsand on the eighth and fourth VAMs, qubits in fifth and sixth VAMs arereleased, polarization rotations of σhd X≡X and σ_(Z)≡Z are performed onthe qubit released from the fifth VAM conditional on the results of theBell measurements on the first and seventh VAMs and on the eighth andfourth VAMs, and similarly, rotations are performed on the photonreleased from the sixth VAM dependent on the results of the Bellmeasurements.
 10. A non-deterministic quantum CNOT gate for photonqubits, comprising an interferometer to receive two target photon modesand two control photon modes and to cause a sign shift by splitting andremixing of the target photon modes, conditional on the presence of aphoton in one particular control photon mode, so that the target photonqubit swaps modes if the control quantum qubit is in one mode but doesnot if the control photon qubit is in the other mode, provided acoincidence is measured between the control and target output photonmodes, where the interferometer comprises five beamsplitters: a firstbeamsplitter to receive two target photon modes, a second beamsplitterto receive one output mode from the first beamsplitter and a controlphoton mode, and to deliver a control output mode, a third beamsplitterto receive the other output mode from the first beamsplitter, a fourthbeamsplitter to receive the outputs of the second and thirdbeamsplitters and deliver the target output modes, and a fifthbeamsplitter to receive the other control photon mode, and deliver theother control output mode.
 11. A non-deterministic quantum CNOT gateaccording to claim 10, where the beamsplitters are asymmetric in phase,reflection off the bottom of the first, third and fourth beamsplittersproduce sign change, but reflections off the top of the second and fifthbeamsplitters produce a sign change.
 12. A non-deterministic quantumCNOT gate according to claim 11, where the first and fourthbeamsplitters are both 50:50 (η=0.5), and the second, third and fourthbeamsplitters have equal reflectivities of one third (η=0.33).
 13. Anon-deterministic quantum CNOT gate according to claim 10, where thefirst and fourth beamsplitters are both 50:50 (η=0.5), and the second,third and fourth beamsplitters have equal reflectivities of one third(η=0.33).
 14. A non-deterministic quantum CNOT gate according to claim10 where the Heisenberg equations relating the control and target inputmodes to the their corresponding outputs are: $\begin{matrix}{c_{Ho} = {\frac{1}{\sqrt{3}}\left( {c_{H} + {\sqrt{2}\mspace{11mu} v_{4}}} \right)}} \\{c_{Vo} = {\frac{1}{\sqrt{3}}\left( {{- c_{V}} + {\sqrt{2}\mspace{11mu} t^{\prime}}} \right)}} \\{t_{Ho} = {\frac{1}{\sqrt{2}}\left( {t^{''} + t^{\prime''}} \right)}} \\{t_{Vo} = {\frac{1}{\sqrt{2}}\left( {t^{''} - t^{''\prime}} \right)}} \\{{where}\text{:}} \\{t^{''} = {\frac{1}{\sqrt{3}}\left( {t^{\prime} + {\sqrt{2}c_{v}}} \right)}} \\{t^{''\prime} = {{\frac{1}{\sqrt{6}}\left( {t_{H} - t_{v}} \right)} + {\sqrt{\frac{2}{3}}\; v_{5}}}} \\{t^{\prime} = {\frac{1}{\sqrt{2}}{\left( {t_{H} + t_{v}} \right)\;.}}}\end{matrix}$
 15. A deterministic CNOT gate comprising anon-deterministic quantum CNOT gate according to claim 10, and furthercomprising two spontaneous down-converters to produce pairs ofpolarization entangled photons which are stored in first, second, thirdand fourth verifiable quantum memorys (VAMs), where two of the storedphotons are released and sent through the non-deterministic quantum CNOTgate and are subsequently stored in fifth and sixth VAMs, andpolarization rotations are performed conditional on the results of Bellmeasurements to achieve the required functionality.
 16. A deterministicCNOT gate according to claim 15, where photons in the second and thirdVAM are released and sent through the non-deterministic quantum CNOTgate, and they are then stored in the fifth and sixth VAMs, and thephotons stored in the first, fifth, sixth and fourth VAMs are in thestate required to implement the teleportation protocol.
 17. Adeterministic CNOT gate according to claim 16, where control and targetphoton qubits enter and are stored in a seventh and eighth VAM, Bellmeasurements are made on the qubits stored in the first and seventh VAMsand on the eighth and fourth VAMs, qubits in fifth and sixth VAMs arereleased, polarization rotations of σ_(X)≡X and σ_(Z)≡Z are performed onthe qubit released from the fifth VAM conditional on the results of theBell measurements on the first and seventh VAMs and on the eighth andfourth VAMs, and similarly, rotations are performed on the photonreleased from the sixth VAM dependent on the results of the Bellmeasurements.